Some varieties containing relation algebras

Three varieties of algebras are introduced which extend the variety RA of relation algebras. They are obtained from RA by weakening the associative law for relative product, and are consequently called nonassociative, weakly-associative and semiassociative relation algebras, or NA, WA, and SA, respectively. Each of these varieties arises naturally in solving various problems concerning relation algebras. We show, for example, that WA is the only one of these varieties which is closed under the formation of complex algebras of atom structures of algebras, and that WA is the closure of the variety of representable RA's under relativization. The paper also contains a study of the elementary theories of these varieties, various representation theorems, and numerous examples. 0. Introduction. Relation algebras (RA's) have a binary operation; which serves as an abstract algebraic analogue of the relative product of binary relations. (The relative product of R, S C U X U, is R I S = {(x, z):(x, y) E R and (y, z) E S for somey E U).) The relative product is associative, and one of the postulates for RA's is that ; is associative. The associativity of relative product can be expressed by a sentence in a first-order language with binary relation symbols, namely ( 1 ) ~VV [3z(3y(Rxy A Syz) A Tzy) -3z(Rxz A 3X(Szx A Txy))]. Although this sentence has three variables, it cannot be proved from the ordinary axioms of first-order logic without using four variables. In contrast, all the other postulates for relation algebras can not only be expressed but proved using only three variables. (These facts were first proved by Tarski. For a proof that (1) requires four variables to prove, see [3].) Tarski asked whether there are any other equations whose translations into first-order sentences can be expressed and proved using only three variables, but which are not derivable from the postulates for RA's without using the associativity of ;. There are such equations. One of them is a special case of the associative law for; called the "semiassociative law", (2) x; (1; 1) = (x; 1); 1 . The class SA of semiassociative relation algebras is defined by the postulates for RA's with the associative law for; replaced by (2). This class properly includes RA, as will be shown in this paper. Received by the editors November 12, 1980. 1980 Mathematics Subject Classification. Primary 03G25, 06E99.